Improved Rates of Bootstrap Approximation for the Operator Norm: A Coordinate-Free Approach

Let Σ̂=1/n∑_i=1^n X_i⊗ X_i denote the sample covariance operator of centered i.i.d. observations X_1,…,X_n in a real separable Hilbert space, and let Σ=𝔼(X_1⊗ X_1). The focus of this paper is to understand how well the bootstrap can approximate the distribution of the operator norm error √(n)Σ̂-Σ_op, in settings where the eigenvalues of Σ decay as λ_j(Σ)≍ j^-2β for some fixed parameter β>1/2. Our main result shows that the bootstrap can approximate the distribution of √(n)Σ̂-Σ_op at a rate of order n^-β-1/2/2β+4+ϵ with respect to the Kolmogorov metric, for any fixed ϵ>0. In particular, this shows that the bootstrap can achieve near n^-1/2 rates in the regime of large β – which substantially improves on previous near n^-1/6 rates in the same regime. In addition to obtaining faster rates, our analysis leverages a fundamentally different perspective based on coordinate-free techniques. Moreover, our result holds in greater generality, and we propose a model that is compatible with both elliptical and Marčenko-Pastur models in high-dimensional Euclidean spaces, which may be of independent interest.